Abstract
An integral of a group G is a group H whose derived group (commutator subgroup) is isomorphic to G. This paper continues the investigation on integrals of groups started in the work [1]. We study:
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A sufficient condition for a bound on the order of an integral for a finite integrable group (Theorem 2.1) and a necessary condition for a group to be integrable (Theorem 3.2).
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The existence of integrals that are p-groups for abelian p-groups, and of nilpotent integrals for all abelian groups (Theorem 4.1).
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Integrals of (finite or infinite) abelian groups, including nilpotent integrals, groups with finite index in some integral, periodic groups, torsion-free groups and finitely generated groups (Section 5).
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The variety of integrals of groups from a given variety, varieties of integrable groups and classes of groups whose integrals (when they exist) still belong to such a class (Sections 6 and 7).
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Integrals of profinite groups and a characterization for integrability for finitely generated profinite centreless groups (Section 8.1).
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Integrals of Cartesian products, which are then used to construct examples of integrable profinite groups without a profinite integral (Section 8.2).
We end the paper with a number of open problems.
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Acknowledgements
The authors would like to thank the referee for a careful reading which led to several improvements of the exposition and proofs.
The first author was funded by national funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 (https://doi.org/10.54499/UIDB/00297/2020) and UIDP/00297/2020 (https://doi.org/10.54499/UIDP/00297/2020)(Center for Mathematics and Applications).
The first, second and fourth authors gratefully acknowledge the support of the Fundação para a Ciência e a Tecnologia (CEMAT-Ciências FCT projects UIDB/04621/2020 and UIDP/04621/2020).
The fourth and fifth authors are members of the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA) of the Istituto Nazionale di Alta Matematica (INdAM), and the fourth author gratefully acknowledges the support of the Universit’a degli Studi di Milano—Bicocca (FA project ATE-2017-0035 “Strutture Algebriche”).
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In April 2018, Carlo Casolo sent the other authors detailed answers to some of the questions in the first version of the paper [1], and we immediately invited him to join us. He was very dedicated and curious about integrals and inverse group theory problems. In fact, the current paper is in large part Carlo’s work, together with the fruits of a meeting in Florence in February 2020. Carlo passed away not long after. He was very generous and kind to all of us and is sorely missed. We dedicate this paper to his memory.
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Araújo, J., Cameron, P.J., Casolo, C. et al. Integrals of groups. II. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2610-4
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DOI: https://doi.org/10.1007/s11856-024-2610-4